Simplify and expand the following expression: $ \dfrac{a}{3a - 10}+\dfrac{3a}{a + 7} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3a - 10)(a + 7)$ Multiply the first term by $\dfrac{a + 7}{a + 7}$ $ \begin{align*} \dfrac{a}{3a - 10} \times \dfrac{a + 7}{a + 7} & = \dfrac{(a)(a + 7)}{(3a - 10)(a + 7)} \\ & = \dfrac{a^2 + 7a}{(3a - 10)(a + 7)}\end{align*} $ Multiply the second term by $\dfrac{3a - 10}{3a - 10}$ $ \begin{align*} \dfrac{3a}{a + 7} \times \dfrac{3a - 10}{3a - 10} & = \dfrac{(3a)(3a - 10)}{(a + 7)(3a - 10)} \\ & = \dfrac{9a^2 - 30a}{(a + 7)(3a - 10)}\end{align*} $ Now we have: $ = \dfrac{a^2 + 7a}{(3a - 10)(a + 7)} + \dfrac{9a^2 - 30a}{(a + 7)(3a - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{a^2 + 7a + 9a^2 - 30a}{(3a - 10)(a + 7)} $ $ = \dfrac{10a^2 - 23a}{(3a - 10)(a + 7)}$ Expand the denominator: $ = \dfrac{10a^2 - 23a}{3a^2 + 11a - 70}$